Properties of 2D and 3D Shapes

This lesson covers the properties of common two-dimensional and three-dimensional shapes for AQA GCSE Mathematics. You need to know the properties of quadrilaterals, types of triangles, symmetry, and the features of 3D solids. You should also be able to draw and interpret plans and elevations. These topics underpin many geometry questions and link to area, volume and transformation work.

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Properties of Quadrilaterals

A quadrilateral is any four-sided polygon. You must know the specific properties of each type.

Quadrilateral	Sides	Angles	Diagonals	Lines of Symmetry	Rotational Symmetry
Square	4 equal	4 right angles (90 degrees)	Equal, bisect at 90 degrees	4	Order 4
Rectangle	Opposite sides equal	4 right angles	Equal, bisect each other	2	Order 2
Rhombus	4 equal	Opposite angles equal	Bisect at 90 degrees, unequal length	2	Order 2
Parallelogram	Opposite sides equal and parallel	Opposite angles equal	Bisect each other, unequal	0	Order 2
Trapezium	One pair of parallel sides	Varies	Varies	0 (unless isosceles: 1)	Order 1 (unless isosceles)
Kite	Two pairs of adjacent sides equal	One pair of opposite angles equal	One bisects the other at 90 degrees	1	Order 1

Exam Tip: A common exam question asks you to identify a quadrilateral from a description of its properties. Learn the table above thoroughly — especially the diagonal properties, which many students forget.

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Types of Triangles

Type	Side Properties	Angle Properties	Lines of Symmetry
Equilateral	All 3 sides equal	All angles 60 degrees	3
Isosceles	2 sides equal	2 base angles equal	1
Scalene	No sides equal	No angles equal	0
Right-angled	Contains a right angle	One angle is 90 degrees	0 (unless also isosceles: 1)

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Symmetry

Line Symmetry (Reflective Symmetry)

A shape has a line of symmetry if one half is the mirror image of the other when folded along that line.

Rotational Symmetry

A shape has rotational symmetry of order n if it looks the same n times during a full 360-degree rotation. Every shape has rotational symmetry of at least order 1 (it maps onto itself after a full turn).

graph LR
    A[Symmetry] --> B[Line Symmetry]
    A --> C[Rotational Symmetry]
    B --> B1[Mirror line divides shape into two identical halves]
    C --> C1[Shape maps onto itself when rotated]
    C --> C2[Order = number of times shape matches in 360 degrees]

Worked Example 1

State the order of rotational symmetry and the number of lines of symmetry of a regular hexagon.

Solution:

• A regular hexagon has 6 lines of symmetry (3 through opposite vertices, 3 through midpoints of opposite sides)
• It has rotational symmetry of order 6

Exam Tip: For regular polygons, the number of lines of symmetry and the order of rotational symmetry both equal the number of sides.

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3D Shapes: Key Vocabulary

Term	Definition
Face	A flat surface of a 3D shape
Edge	The line where two faces meet
Vertex (plural: vertices)	A point where edges meet (a corner)
Cross-section	The shape you see when you slice through a 3D shape
Prism	A 3D shape with a uniform cross-section throughout its length

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Common 3D Shapes

Shape	Faces	Edges	Vertices	Notes
Cube	6	12	8	All faces are squares
Cuboid	6	12	8	All faces are rectangles
Triangular prism	5	9	6	Cross-section is a triangle
Cylinder	3 (2 flat, 1 curved)	2	0	Circular cross-section
Cone	2 (1 flat, 1 curved)	1	1 (apex)	Circular base tapering to a point
Sphere	1 (curved)	0	0	Every point on the surface is equidistant from the centre
Square-based pyramid	5	8	5	Square base with 4 triangular faces
Triangular-based pyramid (Tetrahedron)	4	6	4	All faces are triangles
Hexagonal prism	8	18	12	Hexagonal cross-section

Euler's Formula

For any convex polyhedron: Vertices - Edges + Faces = 2

This is written as V - E + F = 2.

Worked Example 2

A pentagonal prism has 7 faces. Verify that Euler's formula holds.

Solution: A pentagonal prism has:

• Faces = 7 (2 pentagons + 5 rectangles)
• Vertices = 10 (5 on top, 5 on bottom)
• Edges = 15 (5 on top, 5 on bottom, 5 connecting)

Check: V - E + F = 10 - 15 + 7 = 2 (Euler's formula confirmed)

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Plans and Elevations

A plan is the view of a 3D shape from directly above.

An elevation is the view from the front or the side.

View	Direction
Plan	From above (bird's eye view)
Front elevation	From directly in front
Side elevation	From directly to the side

Worked Example 3

A solid is made from unit cubes. The front elevation shows a 2x2 square. The side elevation shows an L-shape (2 high on the left, 1 high on the right). The plan shows a 2x2 square.

Describe the solid.

Solution: The solid is made from 6 unit cubes arranged as a 2x2 base layer with two additional cubes stacked on the left side (forming the L-shape seen from the side).

Exam Tip: When drawing plans and elevations, use a ruler and draw to scale. When constructing a 3D shape from its plans and elevations, use isometric paper. Always label which view is which in your answer.

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Prisms and Their Cross-Sections

A prism is a 3D shape with a uniform cross-section. This means that if you slice it at any point along its length, you always get the same shape.

Prism	Cross-Section
Cuboid	Rectangle
Triangular prism	Triangle
Cylinder	Circle
Hexagonal prism	Regular hexagon
Pentagonal prism	Regular pentagon

Important Distinction

• A pyramid is NOT a prism — its cross-section changes as you move from base to apex.
• A cone is NOT a prism for the same reason.
• A sphere is NOT a prism because it has no uniform cross-section along a length.

Exam Tip: If the exam asks "Is shape X a prism? Explain your answer," you must refer to the cross-section. Say: "Yes, because it has a uniform cross-section" or "No, because the cross-section changes along its length."

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Summary